SINGLE-BYTE / BURST ERROR DETECTING SEC-DED CODES in .NET

Produce Quick Response Code in .NET SINGLE-BYTE / BURST ERROR DETECTING SEC-DED CODES
SINGLE-BYTE / BURST ERROR DETECTING SEC-DED CODES
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Theorem 6.2
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[BOSS78] Let H be the R Jb matrix: H M1 Q H1 M2 Q H2 . . . MJ ... ... !
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Q Hj ;
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! Mi Hi ; 1 i Q 3 2 0 7 6 7 60 7 6 6 : I b 1 7 Q 6 7 7 6 5 4: 0
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Q is a b 1 b matrix consisting of an all-0 s column and the identity matrix of dimension b 1, and M i is an R b 1 b matrix whose columns are b copies of the binary representation of integer i. The code expressed as the foregoing H matrix is a single b-bit byte error detecting SEC code (SEC-SbED code) having code length in bits N b 2R b 1 1 . In this chapter the code length in bits N is a multiple of b, whereas the check-bit length R is not always a multiple of b. The reader should be careful not to confuse these with the notations of a previous section. Theorem 6.2 can be easily proved such that the code satis es conditions 1, 2, and 3 in Theorem 6.1. Example 6.1 [BOSS78] have a (28, 22) SEC-S4ED code with an H matrix 3 0000 0000 1111 1111 1111 1111 1111 1111 0000 0000 1111 1111 7 7 7 0000 1111 0000 1111 0000 1111 7 7: 0100 0100 0100 0100 0100 0100 7 7 7 0010 0010 0010 0010 0010 0010 5 0001 0001 0001 0001 0001 0001
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For R 6, and b 4, we 2 0000 6 0000 6 6 6 1111 H 6 6 0100 6 6 4 0010 0001
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The foregoing can be made an SEC-DED-S4ED code by adding an all-1 s row to the matrix. The following matrix expresses a simple example of this type of SED-DED-S4ED code: 2 6 6 0000 6 6 6 0000 H 6 6 0100 6 6 6 0010 4 0001 1111 1111 1111 1111 3 7 0000 1111 1111 7 7 7 1111 0000 1111 7 7: 0100 0100 0100 7 7 7 0010 0010 0010 7 5 0001 0001 0001
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CODES FOR HIGH-SPEED MEMORIES III: BIT / BYTE ERROR CONTROL CODES
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In general, for the SEC-DED-SbED codes, the following condition is necessary in addition to the conditions 1, 2, and 3 of Theorem 6.1: 4. Ei HT Ej HT 6 Ek HT for all Ei , Ej , Ek 2 E1 , i 6 j 6 k 6 i.
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Theorem 6.3 [REDD78] The codes given by the following H matrix are SEC-SbED codes when b 2; 3; or 4. When b ! 5, the codes are SEC-DED-SbED. The code length (in bits) of the codes is N b 2R b 1 1 .
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b(2 R-b+1- 1)
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H= HOE
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HOO where,
b 0 1 1 . . . 1
. . . .
b 1 1 1 . . . 1
HOE =
Ib-1
HOO =
Ib-1
Used for b an even integer
Used for b an odd integer
Hi : (R b + 1) b matrix whose columns are b copies of the binary representation of
integer i, i = 1, 2, . . . , 2
R-b+1
1,
Ib-1 : (b 1) (b 1) identity matrix.
Proof The possible patterns of syndromes corresponding to the single-bit errors and single-byte errors are given in Figure 6.1. The patterns in the gure indicate whether the top R b 1 positions of the syndrome are zero or nonzero and also the number of ones or the actual bit pattern in the last b 1 positions of the syndrome. In the gure note that the syndromes for the byte errors are nonzero and are different from the syndromes for the single-bit errors. Hence, for 2 b 4, the codes simultaneously correct all single-bit errors and detect all single-byte errors. Similarly, for b ! 5, the possible pattern of syndromes corresponding to single-bit errors, doublebit errors, and single-byte errors are given in Figure 6.2. From this gure it is apparent that the syndromes for the single-byte errors and double-bit errors are nonzero and are different from the syndromes for the single-bit errors. Hence, for b ! 5,